As is known, a radio receiver, in particular in a multimedia system of a motor vehicle, is able to receive a radio signal, in particular an FM radio signal, FM being the acronym of “frequency modulation”.
Such an FM radio signal, received in modulated form by a radio receiver, is subjected to various sensors and to suitable filtering so that the corresponding demodulated radio signal is able to be played back under good conditions, in particular in the passenger compartment of a motor vehicle.
Those skilled in the art know the operating principle of an FM, that is to say frequency-modulated, radio signal received by a suitable radio receiver, with a view to being demodulated and then played to listeners.
A known problem that relates to the reception of an FM radio signal via a mobile radio receiver, in particular one incorporated into a motor vehicle, resides in the fact that the FM radio signal emitted by an emitter may be reflected by natural obstacles or buildings for example, before being received by an antenna of the radio receiver. In other words, the emitted radio signal, before being received by an antenna of the receiver, may have followed various paths, of relatively long or short length.
As a result thereof a selectivity is necessary, because a given radio signal may be received by one antenna several times, with various time shifts. This problem is known to those skilled in the art, who generally refer to it as “multi-path”.
With reference to the FIGURE, to partially mitigate the aforementioned drawbacks relative to multi-path, impulse response filters FIR aiming to remove the interference generated by the multiplicity of the received signals because of the multi-path effect, which was described above, have been developed. These filters FIR implement CMA algorithms, SMA being the acronym of “Constant Modulus Algorithm”, that are configured to cancel out, from the set of the received signals yn, those signals corresponding to secondary signals generated by the multi-path effect, with a view to delivering a processed FM radio signal zn.
Thus, again with reference to the FIGURE, xn represents the signal emitted by the emitting antenna, which by definition has a constant modulus, and yn represents the radio signal received by the receiving antenna of the radio receiver of the vehicle in question, C representing the transfer function of the transmission channel between said emitting antenna and said receiving antenna.
The received radio signal yn has a non-constant modulus because it is the result of the combination of a plurality of time-shifted received signals, i.e. signals resulting from various emitted signals xn that are delayed to a greater or lesser extent.
In the end, the processed radio signal zn is the radio signal reconstructed after application of the CMA algorithm.
In the prior art, algorithms for removing multi-path signals are generally of the “constant modulus” type. Specifically, the principle of frequency modulation ensures that the emitted radio signal has a constant modulus. Thus, computational algorithms called CMA algorithms have been developed and those skilled in the art are constantly seeking to improve them, with for main constraint to ensure, after computation, a substantially constant modulus of the radio signal combined within the receiver, after processing.
CMA algorithms are iterative computational algorithms the objective of which is to determine the real and imaginary parts of complex weights to be applied to the FM radio signals received by an antenna of a radio receiver, with a view to combining them, so as to remove from the combined radio signal the interference due to multi-path.
From a mathematical point of view, the principle presented above, in which complex weights are attributed to multiple radio signals received by an antenna of a radio receiver, including in particular signals received after reflection, which are a source of multi-path interference, with a view to forming a combined radio signal to be played, after canceling out the interference due to multi-path, may be expressed as follows.
The combined radio signal is written:
      z    n    =                    W        n        T            ⁢              Y        n              =                            ∑                      k            =            0                                K            -            1                          ⁢                                            w                              (                k                )                                      _                    ⁢                      y                          (                              n                -                k                            )                                          =                        ∑                      k            =            0                                K            -            1                          ⁢                                            (                                                a                                      (                    k                    )                                                  +                                  j                  ⁢                                                                          ⁢                                      b                                          (                      k                      )                                                                                  )                        _                    ⁢                      y                          (                              n                -                k                            )                                          
where, at the time n, yn is the radio signal, in complex baseband, received by the antenna in question and w(n) is the complex weight attributed, via an impulse response filter, to said received radio signal.
In the prior art, CMA algorithms are implemented to determine the complex vector Wn able to minimize the following cost function:HCMA=E{(|zn|2−R2)2},
where R is a constant to be determined, corresponding to the constant modulus of the combined signal.
In the prior art, the vector Wn of complex weights is considered to consist of linear complex numbers, said vector Wn therefore having the following form:
      W    ⁢                  ⁢    n    =      [                                                      a                              (                0                )                                      +                          j              ⁢                                                          ⁢                              b                                  (                  0                  )                                                                                                                    a                              (                1                )                                      +                          j              ⁢                                                          ⁢                              b                                  (                  1                  )                                                                                          ⋮                                      ⋮                                                                a                              (                                  K                  -                  1                                )                                      +                          j              ⁢                                                          ⁢                              b                                  (                                      K                    -                    1                                    )                                                                          ]  
The components of this vector Wn of complex weights, forming the coefficients of an adaptive filter to be applied to the received radio signal, are independent of one another and the real and imaginary parts of each component are also.
The corresponding cost function may be decreased using the instantaneous gradient technique, in order to be written:
                              ∇                      J                          C              ⁢                                                          ⁢              M              ⁢                                                          ⁢              A                                      =                  2          ⁢                      (                                                                                                  z                    n                                                                    2                            -                              R                2                                      )                    ⁢                      ∇                                                                            z                  n                                                            2                                                              =                  2          ⁢                      (                                                                                                  z                    n                                                                    2                            -                              R                2                                      )                    ⁢                      ∇                          (                                                z                  n                                ⁢                                                      z                    _                                    n                                            )                                                              =                  2          ⁢                      (                                                                                                  z                    n                                                                    2                            -                              R                2                                      )                    ⁢                      (                                                            z                  n                                ⁢                                  ∇                                                            z                      _                                        n                                                              +                                                                    z                    _                                    n                                ⁢                                  ∇                                      z                    n                                                                        )                              
With
      ∇          z      n        =                    ∂                  z          n                            ∂                  W          n                      =          [                                                  Y              n                                                                                          -                j                            ⁢                                                          ⁢                              Y                n                                                        ]      and
            ∇                        z          _                n              =                            ∂                                    z              _                        n                                    ∂                      W            n                              =              [                                                                              Y                  n                                _                                                                                                          -                  j                                ⁢                                                                  ⁢                                                      Y                    n                                    _                                                                    ]              ,the following is obtained:
      ∇          J              C        ⁢                                  ⁢        M        ⁢                                  ⁢        A              =      2    ⁢                  (                                                                          z                n                                                    2                    -                      R            2                          )            ⁡              [                                                            2                ⁢                                                                  ⁢                                  Re                  (                                                                                    z                        _                                            n                                        ⁢                                          Y                      n                                                        )                                                                                                        2                ⁢                                                                  ⁢                j                ⁢                                                                  ⁢                                  Im                  (                                                                                    z                        _                                            n                                        ⁢                                          Y                      n                                                        )                                                                    ]            
Namely:∇JCMA=4(|zn|2−R2)znYn 
The way in which the complex weights are updated is therefore expressed by the following formula:Wn+1=Wn−μ(|zn|2−R2)znYn 
A major drawback of known adaptive filtering techniques and CMA algorithms such as they are applied at the present time, with a view to independently determining the complex weights to be applied to the signals received by the antenna of a mobile radio receiver in order to eliminate there from the interference due to multi-path, resides in the fact that they sometimes converge slowly, and above all in the fact that they sometimes converge wrongly. In other words, sometimes complex weights that meet the required conditions lead to a radio signal of poor quality being played.
Stability problems are thus particularly frequent.
As is known to those skilled in the art, this difficulty with rapidly converging to correct and stable solutions is particularly present in the field of FM radio reception, because the only certain constraint exploitable a priori by algorithms resides in the fact that the modulus of the envelope of the frequency-modulated radio signal remains constant.
However, on the other hand, the antenna receives a plurality of radio signals, corresponding to the emitted radio signal having followed various paths, which are either direct or with one or more reflections, and a complex weight must be determined with a view to being applied to each of these radio signals. The equation contains a high number of unknowns and the objective of the CMA algorithms is therefore to determine the best solutions, among a set of non-optimal solutions allowing a constant modulus of the combined radio signal to be ensured.
More particularly, in scenarios where the desired radio signals coexist with radio signals transmitted over adjacent frequency channels, this problem of convergence is more pronounced. It often occurs that the complex weights obtained with CMA algorithms privilege adjacent radio signals to the detriment of the desired radio signals.